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Cayley diagram

C2×GL(2,𝔽3) #982

order96
isoclinismisoclinic class of #226
automorphismAut(G) = D4×S4 (order 192) · Inn(G) = S4 · Out(G) = D4
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

direct productC2×GL(2,𝔽3)

Splittings

FittingC2×Q8 ⋊ D3
derivedSL(2,𝔽3) ⋊ C22

Subgroups

subgroupG/H
center, socle, hypercenterC22S4
commutator, cosocle, hypocenterSL(2,𝔽3)C22
FrattiniC2C2×S4
FittingC2×Q8D3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22

51 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 16

A minimal faithful permutation representation — 2 generators on 10 points:

(3,4,10,5,6,9,8,7);(1,2)(4,7,10)(5,8,9)

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×GL(2,𝔽3), C4⋊GL(2,𝔽3), #5394, …
maximal normal subgroup ofC4×GL(2,𝔽3), C4⋊GL(2,𝔽3), #5394, …
maximal quotient of#5386, C4×GL(2,𝔽3), C4⋊GL(2,𝔽3), …
automorphism group ofC32×C4
outer automorphism group ofC32×C4, #634, C4⋊(C3⋊D3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 189);